It would be a gigantic undertaking requiring manufacture in a reduced gravity environment, likely using either lunar or asteroid(s) for materials to save bringing up material through the Earth's gravity well. Alternatively, if we were to use a comet 'in flight' for construction materials the materials themselves would already have much of the later needed momentum.
The general idea of just hitch-hiking on an interstellar asteroid sounds quite intriguing!
But I see two problems:
1. How do we find one?: With our current telescopes we can barely find medium planets on stable orbits in other solar systems. How do we find tiny asteroids in interstellar space? and even if we find them, how do we predict their courses? Even if we might predict when they come to us it is much harder to predict where they will be going because small uncertainties in trajectory prediction will add up to huge errors on an interstellar scale. Significantly changing the course of these asteroids seems unrealistic to me, so we should know in advance that it will go roughly in the right direction.
2. The time scale: Although interstellar asteroids would have to be incredibly fast to escape the Sun's gravity in the first place, they'd still be very slow in interstellar dimensions. So we'd need a lot of generations on that generational craft. I guess we'd be talking about hundreds of thousands or even millions of years of total travel time. That's a lot of "wasted" generations uselessly consuming resources. Also people tend to get "bad ideas" (see Dante) when they have nothing useful to do. Who knows what technologies might be available in a hundred years.
I have no idea how to do that, but I'm beginning to think that at that scale (even evolution would have to be taken into account), Earth itself should be the spacecraft we propel into another solar system.
But then again, a hundred years from now, things might look totally different than now. Jules Verne wrote about traveling to the Moon about a hundred years before we actually got there. He did what (hard) SciFi authors often do. He extrapolated from the technology he knew. So he sent his astronauts in a bullet that was shot by a giant cannon, because he didn't imagine what the rockets he knew as inferior weapons would evolve into.
The 'ship' will also need a substantial amount of power to generate enough magnetic field to act as a radiation shield - at least until it's out of the Sun's influence.
Actually, you need a magnetic field all the time. There seems to be quite some radiation in intertellar space. The Voyager probes are just transitioning from the Sun's Heliopause into its "Bow Shock", which both are interference effects of the Sun's radiation with some other fields.
Just a short film tip:
Everybody who follows this thread has probably heard of the 2015 film "The Martian". [1] If you liked this film for its realistic approach (as far as SciFi can be realistic), then you might also like the 2013 film "Europa Report" [2] that deals with the manned mission to the Jupiter moon Europa. It gets a bit of an "Outer Limits" touch towards the end, but for the most par it's pretty much "hard" SciFi (for a film).
[1]
https://en.wikipedia.org/wiki/The_Martian_(film)
[2]
https://en.wikipedia.org/wiki/Europa_Report